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Selberg’s central limit theorem for quadratic dirichlet L-functions over function fields

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Abstract

In this article, we study the logarithm of the central value \(L\left( \frac{1}{2}, \chi _D\right) \) in the symplectic family of Dirichlet L-functions associated with the hyperelliptic curve of genus g over a fixed finite field \({\mathbb {F}}_q\) in the limit as \(g\rightarrow \infty \). Unconditionally, we show that the distribution of \(\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |\) is asymptotically bounded above by the full Gaussian distribution of mean \(\frac{1}{2}\log \deg (D)\) and variance \(\log \deg (D)\), and also \(\log \big |L\left( \frac{1}{2}, \chi _D\right) \big |\) is at least \(94.27 \%\) Gaussian distributed. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian distribution.

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Acknowledgements

This work was carried out during the tenure of a NBHM Fellowship (funded by DAE) for the first author at ISI Kolkata, India. This work was carried out during the tenure of an NSERC PDF (funded by the government of Canada) for the second author at the Centre de Recherches Math’ematiques, Montr’eal, QC, Canada. The authors also thank the anonymous referees for their valuable comments and insightful suggestions that have improved the quality of the manuscript.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Pranendu Darbar

  2. York University, Toronto, ON, Canada

    Allysa Lumley

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  1. Pranendu Darbar
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  2. Allysa Lumley
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Correspondence to Allysa Lumley.

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Communicated by Tim Browning.

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Darbar, P., Lumley, A. Selberg’s central limit theorem for quadratic dirichlet L-functions over function fields. Monatsh Math 201, 1027–1058 (2023). https://doi.org/10.1007/s00605-023-01853-y

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